Open, for both questions. In hand: lower bounds $\log k(n)\ge(\log n)^{1/2-o(1)}$ (Erdős 1965, asserted as "not hard to prove" without proof) and, from Erdős's 1976 display (6), $n_k<k^{\log k/\log\log k}$, proved by a short smooth-number count, the bound $\log k(n)\ge(1/\sqrt2-o(1))\sqrt{\log n\log\log n}$ (a substitution made here; the site prints $\log k(n)\gg\sqrt{\log n\log\log n}$), which a forum note of December 2025 (Tang) also proves directly with the same constant; upper bounds $k(n)\le(1+o(1))n^{1/2}$ (a forum argument of October 2025 by Tao, accepted into the site's commentary and checked here) and, reported by Erdős in 1976 without proof, $n_k>k^2\exp((\log k)^c)$, that is $k(n)\le n^{1/2}\exp(-(\log n)^{c'})$; Erdős could not show $k(n)\le n^{1/2-c}$, "a ridiculously weak result". Nothing approaches the displayed question. Erdős's 1976 lower bound, his reported upper bound and Tang's dated note have claim pages: Erdős 1976 (accepted, refereed), Erdős 1976, upper bound (claimed) and Tang 2025 (claimed). Tao's argument is a thread post, not a dated manuscript, so it has no claim page and is recorded below as progress. No later source was found in the search whose scope the Current assessment records; this is a bounded negative finding, not a certificate of openness.